#72328508

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 11364 conjugacy classes of subgroups; only the 3665 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2)(3,4)(5,6,7,8)(10,13)(11,15)(17,18,20,22)(19,21,24,23);(2,3)(5,6,7,8)(9,10,12,13)(11,16,15,14)(17,19,20,24)(18,21,22,23);(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(1,2,4,3)(5,6,7,8)(9,10,12,13)(11,14,15,16)(17,18,20,22)(19,21,24,23);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,22)(21,24);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,24)(22,23)

Polycyclic presentation:

2^10|a^2=h,b^2=ghi,d^2=gh,f^2=g,b^a=bg,c^a=cj,c^b=cgi,d^a=dg,d^b=di,d^c=dj,e^a=eghi,e^b=eghi,e^d=eghi,f^a=fg,f^b=fij,f^e=fgij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328649
maximal normal subgroup of	#72328649

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 2, F.7*F.8*F.9);;
gap> SetPower(coll, 4, F.7*F.8);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);